On Testing and Learning Quantum Junta Channels

Zongbo Bao, Penghui Yao

Abstract:

We consider the problems of testing and learning quantum k-junta channels, which are n-qubit to
n-qubit quantum channels acting non-trivially on at most kout of nqubits and leaving the rest of
qubits unchanged. We show the following.
1. An $\tilde{O}(k)$-query algorithm to distinguish whether the given channel is k-junta channel or is far from
any k-junta channels, and a lower bound $ \Omega(\sqrt{k})$ on the number of queries;
2. An $\tilde{O}(4^k)$-query algorithm to learn a k-junta channel, and a lower bound $\Omega(4^k/k)$ on the number of
queries.
This gives the first junta channel testing and learning results, and partially answers an open problem
raised by Chen et al. (2023). In order to settle these problems, we develop a Fourier analysis frame-
work over the space of superoperators and prove several fundamental properties, which extends the
Fourier analysis over the space of operators introduced in Montanaro and Osborne (2010).